What is the order of a congruence class in a group modulo n?

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In summary, the problem asks to find the order of the congruence class m modulo n, but it seems to be missing some information. However, if we assume that m and n are coprime, we can use Lagrange's Theorem to solve it. The theorem states that the order of any element in the group must divide the order of the group. Therefore, |[m]_n| must divide both n and m, since m\equiv 0 (mod n), and can be written as n\equiv 0 (mod |[m]_n|). This gives us the solution of |[m]_n|=\frac{n}{(m,n)}.
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snakesonawii
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Homework Statement



If m[tex]\in[/tex]Z and [tex]2\leq n\in Z,[/tex] then [tex]|[m]_n|=\frac{n}{(m,n)}[/tex]

Homework Equations



Lagrange's Theorem

The Attempt at a Solution



I am confused simply because it seems like the problem might be missing something. We are asked to find the order of the congruence class m modulo n. But I thought that to even talk about this we must first assume that m and n are coprime. Otherwise we get results like [tex]|[5]_{15}|=\frac{15}{5}=3[/tex]. Yet 5^3=125 which gives you just the class 5 modulo 15 again. If we wanted to look at a cyclic group generated by [tex][5]_{15}[/tex] we would find that it only has two elements, the classes 5 and 10 from repeated multiplication of the class 5, no inverses, and no identity (the congruence class 1 could be an identity but it is never reached by multiplication of 5 to itself).
 
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  • #2
So it seems like this problem is missing something. I think that Lagrange's theorem could be used to solve this problem if the problem was worded a little differently. The theorem states that the order of any element in the group must divide the order of the group. This means that for any element [m]_n in the group \mathbb{Z}_n, |[m]_n| must divide n. This can also be written as n\equiv 0 (mod |[m]_n|). Therefore, |[m]_n| must divide both n and m, since m\equiv 0 (mod n). Thus, |[m]_n|=(m,n).
 

What is an order of congruence classes?

The order of congruence classes refers to the number of distinct equivalence classes in a given congruence relation. It represents the number of unique solutions to a congruence equation.

How is the order of congruence classes calculated?

The order of congruence classes is calculated using the modular arithmetic method. This involves finding the modulus of the congruence relation and then finding all possible remainders when dividing by the modulus. The number of distinct remainders is equal to the order of congruence classes.

Can the order of congruence classes change?

Yes, the order of congruence classes can change depending on the values of the congruence relation. For example, if the modulus is changed, the order of congruence classes will also change. However, the order of congruence classes will remain the same if the congruence relation remains unchanged.

What is the significance of the order of congruence classes?

The order of congruence classes is significant in determining the solvability of a congruence relation. If the order is equal to the modulus, then the congruence relation is said to be solvable. Additionally, the order can also be used in various number theory applications and proofs.

Are there any limitations to the order of congruence classes?

Yes, the order of congruence classes is limited by the size of the modulus. It cannot exceed the modulus, and in some cases, it may be less than the modulus. Additionally, the order may also be affected by the properties of the numbers involved in the congruence relation, such as primality or divisibility.

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